on the genesis of hexagonal shapes

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On the Genesis of Hexagonal Shapes

Tönu Puu

Centre for Regional Science, Umeå University

Working Paper :

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On the Genesis of HexagonalShapes

Tönu Puu

Centre for Regional Science, Umeå University

Font: Minion (Adobe)

Working Paper :

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; Umeå University; - Umeå; SwedenPh.: +--. Fax: +--.

Email: [email protected]/cerum

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On the Genesis of Hexagonal Shapes

Tönu Puu

CERUM, Umeå UniversitySE-90187 Umeå, Sweden

Abstract: Hexagonal shapes for market areas have been dominant in spatial economics ever since Christallerobserved them and Lösch explained their emergence in terms of optimality. It is observed in the following that,though hexagons are best, the differences to for instance squares in terms of efficiency are negligibly small. Asthe existence of hexagonal shapes - in the space economy, as well as in beehives or other patterns of the physicalworld cannot be denied -a different cause for their emergence is proposed. Such an explanation is structuralstability, which allows three market areas, but not four or six, to come together in each common vertex. Thefocus is hence the way market areas are organized in space, rather than their shape.

Introduction

Hexagonal shapes for market areas have been generally accepted ever since Christaller(1933) observed them empirically, and Lösch (1954) explained their emergence in terms oftheir optimality in terms of compactness, hence economy of minimal total transportationcost.

Circular market areas would be the best solution for isolated firms, so the famous rea-soning goes, but in aggregate circles cannot pave the plane or any portion of it. There wouldresult overlaps or empty interstices. When pushed together, deformable circles would changetheir shapes to polygons. Obviously, there cannot be any concave section of the commonboundaries. Any consumers located in the area enclosed by a concave boundary section andits tangent, could be supplied at a lower transportation cost by the firm inside the boundaryrather than by the competitor. Hence, if exclusively convex areas meet, the boundaries mustbe made up of straight line segments. Supposing further that the firms are identical, andhence have market areas of equal size, we are left with a choice among the three regulartessellations by equilateral triangles, squares, or regular hexagons, as well known fromgeometry.

A hexagon has 120 degree angles, so three of them can come together in each corner,making the total of 360. Likewise, four squares with 90 degree angles, or six triangles with60 degree angles again make up 360. But this exhausts the possibilities. The pentagon willnot do, because three of their 108 degree angles only make 324 degrees, so the plane has tobe warped to fit three pentagons. Twelve such pentagons actually make up a regular closedsurface in three dimensions, and polygons with more corners than six cannot even be fittedon a curved surface. All this was known to Kepler, and, as for the plane, even to Greekantiquity. See Coxeter (1969) and Fejes Tóth (1964) for more detail.

Once we have this choice among the three tessellations, it is easy to show that the hexa-gon is more compact and more economical than the square, which in its turn is better thanthe triangle. This is easily acceptable for intuition, because, the hexagon comes closer to theideal circular shape than the square and the triangle. Of course, an octagon would be evenbetter, but, unfortunately, as we know, it is not admitted as a candidate.

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We have to be more explicit as to what compactness means. In the traditional context of theisoperimetric problem, it means maximum area enclosed by a boundary of given length, orminimum boundary length that encloses a given area size. According to legend, QueenDido of Carthage, when defeated, was offered as much land as could be covered by the hideof an ass. Cunningly, she cut it in as a thin a thread as possible, and laid it out in the shape ofa circle. If we had plenty of Didos, competing for space, they would be squeezed together inhexagons. Apart from legend, medieval cities, surrounded by defence walls, often took acircular shape.

Of course, the relation of boundary length to area enclosed is not the same as the relationof transportation cost to area. We have to make this explicit, and, even if the result often isthe same for compactness in terms of minimal boundary length and minimal transportationcost, there are differences. In particular two important issues are involved

i) As soon as we talk of transportation cost in two dimensions, we have to specify whichkind of transportation metric we assume. Given a suitable Manhattan metric, the square iseven more economical than the circle, as we will see below.

ii) Even assuming the Euclidean metric, which Lösch took for granted, the actual advan-tage of a hexagon over a square is much smaller in terms of transportation cost (about 1.4percent saving), than in terms of boundary length (about 7.8 percent saving), which is re-markably small already.

In the sequel we will argue that the actual differences in terms of compactness are sosmall that it is questionable whether hexagons, when they turn up in the real world, shouldbe attributed to optimality. As a matter of fact they turn up with surprising frequency: SeeWeyl (1951), Tromba (1985), and Ball (1999).

The most well known everyday experience is the hive of the honeybee, whose cells arehexagonal in crossection. Darwin attributed this to a "wonderful instinct" which was theultimate evidence for his evolution theory, as it even selected those bees who could best"economize with labour and wax" as those fittest to survive. From biology we also know ofthe radiolarians (Aulonia Hexagona), tiny spherical organisms whose silicon skeletons aremade of hexagonal cells, as illustrated by d'Arcy Wentworth Thompson (1961) in his beau-tiful book.

From physics we know of Bénard's famous experiment, where a shallow viscous liquidin a cylinder is heated from below, and organizes into hexagonal convection cells, where theliquid rises in the centre of each cell and falls at its boundary. Sometimes atmosphere worksas a Bénard liquid, and hexagonal imprint of the winds have been photographed in desertsand. And, from chemistry, similar structures in diffusion have been reported.

The interesting fact is that most of these cases of self organization are in conflict with thenatural boundary conditions. A rectangular frame of the kind we know from beehives couldaccommodate squares only, but not hexagons. There are bound to arise deformed cells alongthe entire boundary. The same is true about Bénard convection - we cannot cover a circlewith hexagons alone.

The case of the radiolarian is even more intriguing: we can build a tetrahedron, or anoctahedron, from four or eight triangles, a cube from six squares, and, as we saw, a dodeca-hedron from twelve pentagons, but we cannot make any closed surface from hexagonsalone!

Not even if we deform the faces, so that they no longer fall flat in the plane, and just keepthe numbers of faces (F), edges (E), and vertices (V), can we achieve this, because the

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combination of these latter numbers results in the wrong Euler-Poincaré index

V + F - E

For each genus of surface it is a given number. For a surface topologically equivalent to theplane it is 1, as we can verify for the hexagon (6+1-6=1), or any number of them. Add, forinstance two more hexagons that meet in a point. We have 3 faces, 15 edges (18-3, threebeing shared), and 13 vertices (18-5, three being shared by two and one by three), so 13+3-15, which is still 1. And so we can continue adding hexagons, without altering the index. Ifwe take any such aggregate of cells drawn on some elastic sheet, and try to wrap it around asphere, just identifying vertices with vertices and edges with edges, we will not succeed.

Consider in stead the cube: It has 6 square faces, 8 vertices, and 12 edges, so the index is8+6-12=2, so this is the Euler-Poincaré index for a simple closed surface topologicallyequivalent to a sphere. We get the same number with four triangles organized as a tetrahe-dron (4+4-6=2), twelve pentagons organized as a dodecahedron (20+12-30=2), or any otherof the Platonic solids, as they are called, but we cannot obtain index 2 with any aggregate ofhexagons. Concerning surface genus and index numbers, see Henle (1979).

It therefore is truly remarkable that hexagons seem to pave the radiolarian, as it for basicmathematical reasons cannot be paved by hexagons, so that it must have some other poly-gon somewhere to make the index equation possible. For instance, a football is made fromalternating hexagonal and pentagonal patches, but the radiolarian is different: It has justhexagons everywhere, there is just a minimum residual number of different patches to pro-vide for the right genus index.

The forces at work must be truly strong to form hexagonal shapes despite adverse bound-ary conditions. Once we see how small the advantage of hexagons in terms of optimality arethe natural question to ask is whether it is likely that hexagons, which undeniably arise, aredue to optimality, or if there may be any other forces at work.

An obvious alternative candidate is stability, more precisely qualitative structural stabil-ity in the sense of structural robustness in the presence of random changes or slightmisspecifications of the model. Indeed, there is not much point in formulating models,whose outcomes are not just slightly changed at such changes or misspecifications, but turninto something completely different. Samuelson (1947) used this idea in formulating his"correspondence principle": An equilibrium would be without interest if it was not stable.As a rule, assuming stability implies a characterization of the structures that are stable, i.e.a gain of information. Similarly, Arnol'd (1983) suggests that, in modelling the physics ofthe pendulum, account should be taken of friction, no matter whether we knew of its exist-ence as an empirical fact or not, just because the frictionless pendulum is structurally unsta-ble model, whereas the damped pendulum is stable.

Another word for structural stability is transversality. The concept was systematicallyexplored in the development of catastrophe theory. It all boils down to a condition for trans-verse intersections of manifolds embedded in a surrounding space: If the sum of the dimen-sions of two intersecting manifolds equals the sum of the dimensions of the intersectionmanifold and of surrounding space, then the intersection is transverse, otherwise it is not.Thus, two curves (1-dimensional) in a plane (2-dimensional) intersect in a point (0-dimen-sional, as 1+1=2+0, but we could hardly fit a third curve to pass the same intersection point,as we would require 1+0=2+? for a transverse intersection. The intersection manifold wouldneed to have negative dimension, and such geometrical objects do not exist.

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Likewise, two surfaces (2-dimensional) in ordinary space (3-dimensional), intersect trans-versely along a curve (1-dimensional), as 2+2=3+1. We can also make a third surface inter-sect this intersection curve in a point, as 2+1=3+0, but, again, we could not make a fourthsurface intersect this point transversely, as we would need 2+0=3+? to be fulfilled.

Hence we see how neatly the transversality principle sums up all that is typical androbust, and discards what is exceptional and occasional. See Poston and Stewart (1978) fordetail.

Let us now again consider the hexagonal tessellation. As we already noted, the tessella-tion has a property apart from the cells having six edges and six vertices. There are alsoexactly three cells meeting in each vertex. If the tessellation were square, four cells wouldmeet in each vertex. Now, the market boundaries are formed where the delivery pricesaccrued with transportation costs of different suppliers break even. Such delivery prices canbe considered as surfaces over geographical space, and hence are two dimensional objectsthat naturally live in three-space. Two such surfaces meet in curves (market boundaries),and three meet at points. But, a fourth delivery price surface could not be fitted to pass thispoint. Hence, three market areas meet transversely, whereas four do not. The case is evenworse with the triangular tessellation, as six areas would have to meet in the vertices.

If we consider market areas of equal size and regular tessellations, then the reverse ofthree cells meeting at the vertices is the hexagonal shape of the cells. Therefore, structuralstability, or transversality, provides a perfect alternative explanation for the hexagonal shapes.

We could easily reconsider Darwin's case. A multitude of bees are building at the sametime, pushing against the walls while making cells of their own size, and the joint actionproduces a shape which is robust. The hexagonal is, but, if the bees had an instinct to buildcells of a square crossection, then the slightest careless move by a bee would deform theconfiguration. Therefore, no recourse to conscious plans or selection of economic instinctsis needed. The cells are hexagonal, just because this configuration is robust.

As a further example, take a look at an atlas of countries with national boundaries: Wealways see three countries coming together in one point, never four or more. The onlyexception is the case of states in federations, where the borders were drawn by pen and rulerin the hands of some administrator, but it would not be robust to perturbations due to wars,negotiations, and peace treaties. Of course, countries are of different size and power, andtherefore we do not see just hexagonal shapes, even topologically. But, given homogeneityand equal size, the reverse of the meeting of three countries would again be hexagons.

This duality between shape and incidence is emphasized by Shäfli's symbols for polygo-nal tessellations {p, q}, where p denotes the number of corners in the polygon, and q de-notes the number of faces that meet in a corner. As

(p- 2)(q - 2) = 4

in the plane, there are just the three possibilities {3, 6}, {4, 4}, and {6, 3}.

Measures of Economy and Compactness

As stated above, there are different measures for how compact or economical a certainpolygonal shape is. Darwin was concerned with the length of boundary for a cell of given

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θπ / n

2π / n

ρR

Figure 1. Illustration to the calculation of side length, area, and transportation volume for apolygon.

area content, i.e., with the classical isoperimetric problem. Lösch focused upon total dis-tance from the centre to all the different points of the polygon, or rather again total distancefrom the respective centres for an aggregate of cells. These measures are, of course, differ-ent, though they result in the same conclusion that the hexagon is better than the square, andthe square better than the triangle. Though both Darwin and Lösch were concerned withcell aggregates, it is easiest to deal with the measures for an individual cell.

To save labour, we can derive the formulas for a general n-gon in one step. Suppose theradius of the circle inscribing the polygon is denoted R. Obviously, we can derive all themeasures we want by taking one of the identical triangular slices with the centre angle

2π / n , and then multiply the measures by n. See Fig. 1.Or, we can do even better: These isosceles triangles have a symmetry axis, so we need

only take one half of these, with the central angle π / n , and finally multiply the measuresby 2n. As the triangle has one right angle, we right away get the contribution to the perim-

eter as R nsinπ , and the contribution to the area as 12

2R n nsin cosπ π . Multiplying by 2n, we

get:

L Rn n= 2 sinπ

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and

A R n n n= 2 sin cosπ π

where L alludes to length (or linear) and A alludes to area. Let us check what happens when

we let n → ∞ . We then have lim sinn nn→∞ =π πb g and further lim sin cosn n nn→∞ =π π πb g ,

so L R= 2π and A R= π 2 , which are the familiar formulas for the perimeter and the area ofa circle, i.e. a polygon with infinitely many sides.

It is slightly more tricky to get total distance from the centre, located at the central angle.We need to integrate:

V n r dr d nRd

R

n

nn n

= =zz z2 22

00

3 33

0

cos /cos

coscos

πθ

π π

θ θθ

π

b g

A word of explanation may be needed. We assume an Euclidean distance metric, so theintegrand r is the straight line distance from the centre. The second power comes from the

change to polar coordinates dy dx r d dr= θ . If we denote the length from the centre in

direction θ by ρ , then we have ρ θ πcos cos= R n , so, solving for ρ , gives us the upper

bound for the inner integration.The expression has a closed form solution:

V nR

n n n n= + +3

34 23

cos sin cos ln tanπ π π π πb gc h

We understand why V for transportation distance alludes to volume, as it is proportional tothe cube of the radius, quite as the area A was proportionate to the square, and the length Ljust proportional. Geometrically V indeed is a volume: If we erect a right prism of unitheight on the polygon, and circumscribe a cone on the same polygonal base, turning itupside down with its centre at the centre of the polygon, then V is the volume of the prismthat lies under the inverted cone. See Fig. 2.

It might be tempting to proceed right away to comparing the L:s for the isoperimetricproblems, and the V:s for transportation, but the outcome would be misleading. First, wehave to calculate R for each polygon so as to make the area unitary. Otherwise boundarylength or transportation distance would just be larger for a larger area. Obviously we hencecannot choose the same radius for all polygons. From the expression for A, we get:

Rn n n

= 1

sin cosπ π

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as the radius that makes the area of that particular polygon unitary. It is this expression wehave to substitute in the expressions for L and V in order to get the measures we want.

Hence, finally:

L n n= 2 tanπ

and

Vn n n

n n n= + +1

3

11 4 2

cos sincos cot ln tan

π ππ π π πb gc h

This expression looks quite messy, but for n ≥ 3, which is fulfilled when we really dealwith a polygon, its graph looks very simple and is a uniformly decreasing function, which

goes asymptotically to 23 / π as n goes to infinity. The same is true for the much simpler

looking L, which goes to the asymptotic value 2 π .Below we list the exact expressions and the numerical approximations for the triangle,

the square, the hexagon, and, finally, the circle:

Figure 2. Transportation distance a the volume of a prism under an inverted cone.

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L3 2 3 4 559034= ⋅ ≈ . V3

19

13

2 2 2 2 0 403614= + + ≈ln .d ie j

L4 4= V416 2 1 2 0 3826= + + ≈ln .d ie j

L6 2 12 3722414= ⋅ ≈ . V6

13

13

1412 3 0 3772

14= + ≈ln .b gc h

L∞ = ≈2 35449π . V∞ = ≈23

1 0 3761π

.

As we see, both in terms boundary length, and in terms of total distance, it is true that thecircular shape is the most economical, and that, among the polygonal shapes in a planetessellation the hexagon is best, the square next best, and the triangle the worst solution.

However, we also see that the differences are very small. In terms of boundary length,the hexagon means a waste of mere 5 percent, and the square of another 7.8 percent. As theideal circular form is impossible in a tessellation, Darwin's honey bees would save 7.8percent as compared to other bees that insist on building cells of square crossection.

The case is even less convincing in Lösch's case of minimizing transportation cost. Theloss due to the fact that the ideal circular market shape is impossible and the hexagon thebest possible is now about 0.3 percent. A transformation from the hexagonal to the squaremarket shape would mean another loss of 1.4 percent. The question to pose, in view ofrelocation costs and other frictions, is whether the saving of 1.4 percent would provide aconvincing explanation for why an actual square tessellation of market areas would betransformed into a hexagonal one.

Too often we just pose questions about optimality, we should also keep in mind howmuch better the optimal solutions are than other equally possible solutions. It might well bethat differences are really small, so that frictions of various kinds prevent the systems fromattaining the ultimate optimal state. Or, in case we actually observe the optimal configura-tions, we should ask if they should not be ascribed to other causes than optimality.

A Case in Solid Geometry

Solid geometry, which we touched upon in the introduction, is, of course, much more com-plicated than plane geometry. The regular convex polyhedra, or Platonic solids, three-di-mensional equivalents of the regular polygons, which we touched upon, are five: The tetra-hedron (with four triangular faces), the cube (with six square faces), the octahedron (witheight triangular faces), the dodecahedron (with twelve pentagonal faces), and the icosahedron(with twenty triangular faces).

The five Platonic solids have fascinated imaginative minds over the ages. Leonardo daVinci made space models of them, and Kepler, once he was caught by the Copernicanheliocentric cosmology, first tried to organize the planetary orbits on spheres which wereinscribed in, and themselves inscribed these Platonic polyhedra. See Fig. 3. He was particu-larly satisfied by the fact that the number of the then known planets agreed with the number

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Figure 3. Kepler's first cosmology, with nested spheres and regular polyhedra.

of concentric spheres needed. But this beautiful theory did only fit the orbits for two planetssufficiently well, so Kepler abandoned the theory in favour of the elliptic orbits which madehim famous.

Soap bubbles are spherical, shaped by minimizing surface tension, and are the simplestinstances of the celebrated Plateau problem. However, none of the space filling convexpolyhedra have any optimality properties, such as the hexagon in the plane tessellation has.In this case it would be maximum compactness in terms of least surface area enclosing agiven volume. As a matter of fact not all the Platonic solids are even space filling. The cubeobviously works, but the icosahedron does not, no matter how we stack them there remaingaps.

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A favourite candidate for such optimality has in stead been the rhombic dodecahedron,which is not a regular solid. It is the shape of the garnet crystal, and has twelve rhombic, notpentagonal faces. It also turns up in beehives, more precisely in the shape the two layers ofcomb cells that stand back to back in each frame. Darwin could have taken this as additionalproof, provided it was certain that the rhombic dodecahedron indeed was the optimal shapein three dimensions. See Fig. 4.

Additional evidence could be ascribed to the following experiment. Soft lead shot, ap-proximately spherical and of equal volume, are loaded in a massive cylinder, and then com-pressed by a piston (letting gunpowder explode behind the piston). The lead shot are de-formed from the spherical shape to the rhombic dodecahedral in each such experiment.

Yet, in his famous Baltimore lectures 1887, Lord Kelvin demonstrated that there existsanother space filling solid which has a better economy of surface area to enclosed volume.This is called the tetrakaidekahedron, and is obtained by cutting off all the six corners of anoctahedron, so as to produce a solid with alternating squares (at the corners) and hexagons(deformed from the original triangular faces). Kelvin showed how the faces and edges fur-ther have to be slightly warped so as to produce the better candidate for optimality. Thisstory is told by Weyl. According to him, nobody knows which is the optimal space fillingsolid, but there is no doubt that Kelvin's tetrakaidekahedra represent a better solution thanthe rhombic dodecahedra. However, neither the tetrakaidekhedron nor any other shapethan the rhombic dodecahedron ever turns up in experiment. This obviously is a case wherefrictions impede that the optimal solution is attained, even though the compression is anoptimizing procedure.

Though we need not be concerned with the phenomena in tree dimensions here, this caseprovides an illustration to the weakness of optimization when differences in objective ful-filment are small.

Other Metrics

Let us, however, consider that we instead of the Euclidean, have a Manhattan type of trans-portation system, with roads in the diagonal, South-West to North-East, and South-East toNorth-West directions, infinitely dense, quite as in the case of the Euclidean metric. Sup-pose further that we have a firm located at the origin.

Then the distance metric from the origin to any point x, y in the plane reads:

x y x y+ + −

Suppose further that the firm has a square market area defined by:

x y x y+ + − ≤ 1

This is tilted 45 degrees as compared to the road directions. The boundary of the square is

the union of four segments of the lines x y= ± = ±05 05. , . . As the square has unit sides itsarea is unity. Suppose the market we consider is part of an infinite tessellation

x y i ji j, ,d i b g= with i, j ranging over all integers. Then, assuming that all firms charge the

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same mill price and have the same transportation cost as given by our cost metric, themarket boundaries are indeed as assumed.

Let us so calculate total transportation cost, for simplicity assuming unit transportationcost, i.e., that the cost has been used as measuring stick to determine the unit distance. Itbecomes:

x y x y dy dx+ + − =−−zz c h0 5

0 5

0 5

0 5 2

3.

.

.

.

For comparison, let us also calculate the cost for a circular market area. But if we take unitradius we will arrive at the area π , which is much larger than the unit area for the square.

So, to make a fair comparison, we have to choose a radius 1/ π , which results in unit

area. The total transportation cost is then:

Figure 4. The rhombic dodecahedron.

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r d dr2

0

2

0

1 8 2

3cos sin cos sin

/

θ θ θ θ θπ π

ππ

+ + − =zz c h

Note that it was convenient to use polar coordinates x r y r= =cos , sinθ θ . We factored

one power of r out from the absolute signs as it is positive in the integration. The second

power arises from the conversion to polar coordinates dy dx r d dr= θ .

This expression is slightly higher than for the square, 0.677... as compared to 0.666... Hence,even the circular area is less economical from the view of transportation cost than the squarearea. Again, differences are small.

Of course, there are many other transportation systems, all leading to different cost metrics,and resulting in different rankings for the shapes of market areas. The Manhattan metric,

i.e. x y+ , if we align the roads in the West-to-East and South-to-North directions, by

rotating them 45 degrees, as compared to the above, and the common Euclidean metric

x y2 2+ , are both examples of the family of Minkowski metrics:

Figure 5. Transportation volume when the heagonal tessellation goes with a correspondingmetric..

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x yα α α

+e j1/

with α = 1 and α = 2 respectively. Dual to each transportation system is a system of iso-distance loci, concentric squares for the Manhattan, and circles for the Euclidean.

The former is most typical for city regions, along with the ring-radial, which has a morecomplicated metric, whereas the Euclidean is the traditional distance concept from geom-etry and used by all the classics. We can also think of roads in three directions inclined at 60degree angles to each other, as particularly convenient for hexagonal cell aggregates, result-ing in the metric:

x y x y x+ + − +3 3 2

Its iso-distance loci are concentric hexagons. This metric would always single out a suitablelayout of hexagons as the best tessellation in terms of least transportation cost, as comparedto all other polygonal tessellations, and even to the circular cell shape. Without presentingthe calculations we just show a picture showing the volume of transportation when such ametric goes with a hexagonal tessellation. See Fig. 5. This is a companion to Fig. 2 wherethe circular cone is replaced by a hexagonal pyramid, again turned upside down.

So, we see how the optimality depends on the distance metric, and conclude that we, inview of the tiny compactness differences between different cell shapes, should be evenmore careful when we are not certain about which distance metric to apply, as a differentchoice reverses the ranking.

Boundary Problems

When hexagonal cells are enclosed in the rectangular frame of a beehive, or the circularvessel in which Bénard convection is illustrated, or even forced to tile the surface of aspherical radiolarian, the boundary influences the optimality properties of the aggregate.The measure of average boundary per cell depends on how many cells there are, and there isalso a waste of space in deformed boundary cells if the shape does not fit exactly in theframe or boundary, as always is the case with hexagons.

As a rule, however, the influence of the boundary decreases the more cells there are inthe aggregate. In general we can say that, in the limit, every polygonal tessellation of what-ever polygonal shape is just half of that for an isolated polygon of the kind. This is sobecause, except for the boundary layer, all edges are shared by two cells. The larger theaggregate is, the smaller is the portion of it that belongs to the boundary layer, and in thelimit its importance vanishes. Hence, given a sizeable cell aggregate, the measures for bound-ary length of the individual cells, standardized to unit area are just halved, and hence theoptimality comparisons still hold. Probably the numbers of beehive cells in a frame or con-vection cells in a vessel are large enough to make the influence of the boundary negligible.

More important for us, however, is that the measures of transportation volume are notaffected by these boundary matters at all. They are matters of the individual cell and itsshape, and independent of how they aggregate together.

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Transversality

The point of the above reasoning has been to show how weak the argument that hexagonalshapes arise due to their optimality in view of their maximum compactness is. On the otherhand, there is no doubt that many hexagonal patterns in fact can be observed in reality. So,we are not denying their existence, but want to find a stronger argument for them.

In the introduction we already indicated one such argument, the structural stability ofhexagonal tessellations, as formalized by the principle of transversality.

To make things precise suppose we have a number of firms at different locations x yi i,b g ,

charging mill prices pi . These prices accrue with accumulated transportation costs along

the radiating optimal routes from the mill locations, and define local consumers' prices atany point (x, y) of the region considered. The choice of optimal routing may be a compli-cated problem, quite like the derivation of the resulting local price. See Puu (2003) forexamples. But, no matter how complex the transportation system is, for instance includingseveral combined modes, the resulting distance metric (with transportation cost absorbed)between any pair of points is always a unique, well defined, function.

Hence, for the sake of generality, denoting the distance function d x x y yi i− −,b g , we

have the local prices:

f x y p d x x y yi i i i, ,b g b g= + − −

If, for instance, we deal with the Euclidean metric, we have

f x y p x x y yi i i i,b g b g b g= + − + −2 2

Obviously each local price function

z f x yi= ,b g

defines a two dimensional surface in three dimensional (x, y, z)-space. Market boundariesare defined through letting two such surfaces intersect. According to the transversality prin-ciple, they intersect along a curve, defined by:

z f x y f x y= =1 2, ,b g b g

For the Euclidean metric:

z p x x y y p x x y y= + − + − = + − + −1 1

2

1

2

2 2

2

2

2b g b g b g b g

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where the two last equations result in the famous implicit quartic function for the marketboundary known from Launhardt's "funnel" construction. The transversality condition is

fulfilled, as the dimensions of the two 2-dimensional surfaces z f x y= 1 ,b g and z f x y= 2 ,b g ,

produce the 1-dimensional intersection in 3-space, so 2+2=3+1.

Consider now a third firm with its supply price surface z f x y= 3 ,b g . Putting:

z f x y f x y f x y= = =1 2 3, , ,b g b g b g

determines a unique point x y z, ,b g . This 0-dimensional point is the intersection of a new 2-

dimensional surface and a 1-dimensional curve in 3-space, so 2+1=3+0. Again, we have atransverse intersection.

However, adding a fourth firm, we would require a non-transverse intersection between

a 0-dimensional point x y z, ,b g and a new 2-dimensional surface z f x y= 4 ,b g , which is

impossible. It would need a very special configuration of its parameters, mill price p4 and

location coordinates x y4 4,b g which would be unlikely. The corresponding parameters for

the three previous firms were part in the determination of the coordinates of the intersection

Figure 6. Three intersecting Launhardt funnels, market boundaries, and the three market point.

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point, but it is now given, and the parameters for the fourth firm had no part in its determi-nation, they would just have to be required to fit.

By conclusion, just three firms and no more meet transversely at each vertex. The sameholds true all over the tessellation, and produces a set of three-market junctions, which arejoined by curve segments which together determine the market areas.

If we add assumptions of convexity, regularity, and equality of market areas, the reverseof this is the hexagonal tessellation. So, it is the only regular tessellation which is robust,and structurally stable. As a conclusion, stability provides a better argument for hexagonsthan does optimality.

Flows and Stability

By conclusion, we should add that it matters to what objects in spatial economics we applystructural stability. We applied the considerations to market areas, cell aggregates, assum-ing a grainy structure of locally monopolistic market areas.

We could, however, also have applied structural stability to flows, of, for instance, tradedcommodities. The outcome is then quite different. We will outline it briefly in what follows,but the reader can find much more material in Puu (2003).

The facts for structural stability of flows in two dimensions are mainly due to M.M.Peixoto. See Peixoto (1977). The characterization of structurally stable plane flows im-plies:

i) There are only a finite number of singular (stagnation) points, or limit cycles;ii) Each of the singularities is of the same type as in linear systems, i.e., a node (stable or

unstable), a focus (again stable or unstable), or a saddle point;iii) There are no so called heteroclinic trajectories connecting saddle points.

Making the further obvious restriction to cases where the flow pattern is an optimizing one,i.e. minimizing transportation cost, eliminates the limit cycles and the foci, and still simpli-fies the issue: We are left with just nodes, stable and unstable, or sinks and sources, andsaddle points, with no trajectories between saddle points.

Now, any flow outside the singular stagnation points (where several trajectories areincident), is topologically equivalent to a set of parallel straight lines, so the singular pointsprovide the organizing element. If we just draw a graph of the set of singular points and

Figure 7. The tessellation elements for flows, basic triangle bright.

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their connecting trajectories, we find that it can always be triangulated. Further, each suchtriangle must have exactly one source, one sink, and one saddle point as vertices. Weobviously cannot have two sources or two sinks, because on a triangle all vertices are con-nected. As a trajectory runs from a source to a sink, we could not orient the direction alongthe side connecting two sinks or two sources. Further, saddle connections were excluded, sowe cannot have two saddles either. This leaves us with only one basic triangle, with onesource, one sink, and one saddle point. The interior of such a triangle could be filled byother flow lines by free hand. They all issue at the source, pass close by the saddle, and endup at the sink.

To arrive at a regular tessellation element, these elementary triangles have to be re-flected in pairs, and arranged cyclically around a singular point of one kind (it makes nodifference which). Taking six, we get a triangular element, taking eight, we get a squareone, and taking twelve, we get a hexagonal one. These can be multiplied, translated, andagain organized in a regular tessellation, this time of the flow pattern. See Fig. 7.

The basic pattern now is square, with equal numbers of sources and sinks, or mixedhexagonal/triangular, with the number of one sort of nodes twice the number of the other.The exclusively hexagonal flow pattern is structurally unstable, because it would have to beorganized around monkey saddles, which are about the most unstable feature we couldthink of.

Though the conclusions are different when transversality is applied to flows than thosewhen they are applied to market areas, we should note that there is no contradiction. Thecase of market areas with central firms and continuously smeared out consumers is a case ofmonopolistic competition. Whenever we focus on flow patterns, firms and consumers areequivalent, i.e., both smeared out in space. Hence, the case is one of a competitive market.That conclusions are different for monopoly and competition should be no surprise.

References

V. I. Arnol'd, 1983, Geometrical Methods in the Theory of Ordinary Differential Equations,(Springer-Verlag)

P. Ball, 1999, The Self-Made Tapestry, (Oxford University Press)W. Christaller, 1933, Die zentralen Orte in Süddeutschland (Jena)H. S. M. Coxeter, 1969, Introduction to Geometry, (John Wiley & Sons, Inc.)M. Henle, 1979, A Combinatorial Introduction to Topology, (Freeman)L. Fejes Tóth, 1964, Regular Figures, (Pergamon Press)A. Lösch, 1954 (translation from the German original 1939, 1943), The Economics of Lo-

cationM. M. Peixoto, 1977, "Generic properties of ordinary differential equations", J. Hale (Ed.),

Studies in Ordinary Differential Equations, (MAA Studie in Mathematics 14: 54-92)T. Poston and I. S. Stewart, 1978, Catastrophe Theory and its Applications, (Pitman)T. Puu, 2001, Mathematical Location and Land Use Theory, (Springer-Verlag)P. A. Samuelson, 1947, Foundations of Economic Analysis (Harvard University Press)A. Tromba, 1985, Mathematics and Optimal Form, (Scientific American Books, Inc.)D. Wentworth Thompson, 1961 (abridged edition from the original 1917, 1942), On Growth

and Form, (Cambridge Univesity Press)H. Weyl, 1952, Symmetry, (Princeton University Press)

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